Part I. Section I. Chapter 6. “Of the Properties of Integers, with respect to their Divisors.”

58 As we have seen that some numbers are divisible by certain divisors, while others are not so; it will be proper, in order that we may obtain a more particular knowledge of numbers, that this difference should be carefully observed, both by distinguishing the numbers that are divisible by divisors from those which are not, and by considering the remainder that is left in the division of the latter. For this purpose let us examine the divisors;

2, 3, 4, 5, 6, 7, 8, 9, 10, etc.

59 First let the divisor be 2; the numbers divisible by it are, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, etc. which, it appears, increase always by two. These numbers, as far as they can be continued, are called even numbers. But there are other numbers, namely

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, etc.

which are uniformly less or greater than the former by unity, and which cannot be divided by 2, without the remainder 1; these are called odd numbers.

The even numbers are all comprehended in the general expression \(2a\); for they are all obtained by successively substituting for \(a\) the integers 1, 2, 3, 4, 5, 6, 7, etc. and hence it follows that the odd numbers are all comprehended in the expression \(2a + 1\), because \(2a + 1\) is greater by unity than the even number \(2a\).

60 In the second place, let the number 3 be the divisor; the numbers divisible by it are,

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, and so on;

which numbers may be represented by the expression \(3a\); for \(3a\), divided by 3, gives the quotient a without a remainder. All other numbers which we would divide by 3, will give 1 or 2 for a remainder, and are consequently of two kinds. Those which after the division leave the remainder 1, are,

1, 4, 7, 10, 13, 16, 19, etc.

and are contained in the expression \(3a + 1\); but the other kind, where the numbers give the remainder 2, are,

2, 5, 8, 11, 14, 17, 20, etc.

which may be generally represented by \(3a + 2\); so that all numbers may be expressed either by \(3a\), or by \(3a + 1\), or by \(3a + 2\).

61 Let us now suppose that 4 is the divisor under consideration; then the numbers which it divides are,

4, 8, 12, 16, 20, 24, etc.

which increase uniformly by 4, and are comprehended in the expression \(4a\). All other numbers, that is, those which are not divisible by 4, may either leave the remainder 1, or be greater than the former by 1; as,

1, 5, 9, 13, 17, 21, 25, etc.

and consequently may be comprehended in the expression \(4a+1\): or they may give the remainder 2; as,

2, 6, 10, 14, 18, 22, 26, etc.

and be expressed by \(4a+2\); or, lastly, they may give the remainder 3; as,

3, 7, 11, 15, 19, 23, 27, etc.

and may then be represented by the expression \(4a+3\). All possible integer numbers are contained therefore in one or other of these four expressions;

\[4a, \; 4a + 1, \; 4a+ 2, \; 4a + 3.\]

62 It is also nearly the same when the divisor is 5; for all numbers which can be divided by it are comprehended in the expression \(5a\), and those which cannot be divided by 5, are reducible to one of the following expressions:

\[5a+1, \; 5a+2, \; 5a + 3, \; 5a + 4;\]

and in the same manner ve may continue, and consider any greater divisor.

63 It is here proper to recollect what has been already said on the resolution of numbers into their simple factors; for every number, among the factors of which is found 2, or 3, or 4, or 5, or 7, or any other number, will be divisible by those numbers. For example; 60 being equal to 2 · 2 · 3 · 5, it is evident that 60 is divisible by 2, and by 3, and by 5.

64 Farther, as the general expression \(abcd\) is not only divisible by \(a\), and \(b\), and \(c\), and \(d\), but also by

\(ab\), \(ac\), \(ad\), \(bc\), \(bd\), \(cd\) and by
\(abc\), \(abd\), \(acd\), \(bcd\), and lastly by
\(abcd\), that is to say, its own value;

it follows that 60, or 2 · 2 · 3 · 5, may be divided not only by these simple numbers, but also by those which are composed of any two of three; that is to say, by 4, 6, 10, 15: and also by those which are composed of any three of its simple factors; that is to say, by 12, 20, 30, and lastly also, by 60 itself.

65 When, therefore, we have represented any number, assumed at pleasure, by its simple factors, it will be very easy to exhibit all the numbers by which it is divisible. For we have only, first, to take the simple factors one by one, and then to multiply them together two by two, three by three, four by four, etc. till we arrive at the number proposed.

66 It must here be particularly observed, that every number is divisible by 1; and also, that every number is divisible by itself; so that every number has at least two factors, or divisors, the number itself, and unity: but every number which has no other divisor than these two, belongs to the class of numbers, which we have before called simple, or prime numbers.

Except these simple numbers, all other numbers have, beside unity and themselves, other divisors, as may be seen from the following Table, in which are placed under each number all its divisors.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 3 2 5 2 7 2 3 2 11 2 13 2 3 2 17 2 19 2
4 3 4 9 5 3 7 5 4 3 4
6 8 10 4 14 15 8 6 5
6 16 9 10
12 18 20
1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5 2 6 2 6

67 Lastly, it ought to be observed that 0, or nothing; may be considered as a number which has the property of being divisible by all possible numbers; because by whatever number \(a\) we divide 0, the quotient is always 0; for it must be remarked, that the multiplication of any number by nothing produces nothing, and therefore 0 times \(a\), or \(0a\), is 0.

Editions

  1. Leonhard Euler. Elements of Algebra. Translated by Rev. John Hewlett. Third Edition. Longmans, Hurst, Rees, Orme, and Co. London. 1822.
  2. Leonhard Euler. Vollständige Anleitung zur Algebra. Mit den Zusätzen von Joseph Louis Lagrange. Herausgegeben von Heinrich Weber. B. G. Teubner. Leipzig and Berlin. 1911. Leonhardi Euleri Opera omnia. Series prima. Opera mathematica. Volumen primum.